Euclidean geometry is a common example of this: if we remove the parallel postulate, we get a weaker theory; unable to prove various "facts". However, this weak

Euclidean geometry is a common example of this: if we remove the parallel postulate, we get a weaker theory; unable to prove various "facts". However, this weaker theory is also more general, applying to curved spaces too; where those facts are not true. It turns out that the physical universe is curved, and hence those original "facts" don't hold in the 'real world'. It seems like a similar thing is happening here: we can prove that the set of reals is uncountable, but only if we use classical axioms which don't account for computability. Using a weaker (intuitionistic/constructive) theory prevents us proving that the reals are uncountable; that turns out to be necessary, because some of the alternatives cause the reals to be countable. Since the physical universe is computable (or at least, computably indistinguishable from being computable), these alternatives may be better descriptions of the 'real world' than classical logic.